Systematic approach to conformal systems with extended Virasoro symmetries

The Toda field theories, which exist for every simple Lie group, are shown to give realizations of extended Virasoro algebras that involve generators of spins higher than or equal to two. They are uniquely determined from the canonical lagrangian formalism. The quantization of the Toda field theories gives a systematic treatment of generalized conformal bosonic models. The well-known pattern of conformal field theories with non-extended Virasoro algebra, appears to be repeated for any simple group, leading to a “periodic table”, parallel to the mathematical classification of simple Lie groups.     

Born's rule from classical random fields

This Letter is an attempt to go beyond QM. In our approach density operators of QM can be represented as covariance operators of classical random fields. Born's rule can be obtained from measurement theory for classical random field under the assumption that the probability of detection of field is proportional to the power of this field.     

Hidden global conformal symmetry without Virasoro extension in theory of elasticity

The theory of elasticity (a.k.a. Riva–Cardy model) has been regarded as an example of scale invariant but not conformal field theories. We argue that in d=2$d=2$ dimensions, the theory has hidden global conformal symmetry of SL(2,R)×SL(2,R)$SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ without its Virasoro extension. More precisely, we can embed all the correlation functions of the displacement vector into a global conformal field theory with four-derivative action in terms of two scalar potential variables, which necessarily violates the reflection positivity. The energy–momentum tensor for the potential variables cannot be improved to become traceless so that it does not show the Virasoro symmetry even with the existence of global special conformal current.     

Deformed fields and Moyal construction of deformed super Virasoro algebra

Studied is the deformation of super Virasoro algebra proposed by Belov and Chaltikian. Starting from abstract realizations in terms of the FFZ type generators, various connections of them to other realizations are shown, especially to deformed field representations, whose bosonic part generator is recently reported as a deformed string theory on a noncommutative world-sheet. The deformed Virasoro generators can also be expressed in terms of ordinary free fields in a highly nontrivial way.     

Construction of classical Virasoro algebras as SU(1,1) extensions

The generators of SU(1,1) and their Casimir invariant may be arrayed in a compact formula to construct all generators of the centreless Virasoro algebra which includes this SU(1,1) subalgebra. We illustrate properties of our construction through a simple differential operator realization of these algebras and comment on its usefulness.     

Orbifold conformal blocks and the stack of pointed G-covers

Starting with a vertex algebra V, a finite group G of automorphisms of V, and a suitable collection of twisted V-modules, we construct (twisted) D-modules on the stack of pointed G-covers, introduced by Jarvis, Kaufmann, and Kimura. The fibers of these sheaves are spaces of orbifold conformal blocks defined in joint work with Edward Frenkel. The key ingredient is a G-equivariant version of the Virasoro uniformization theorem.     

Cohomology of the Virasoro algebra with coefficients in string fields

The first cohomology of the Virasoro algebra with coefficients in string fields are investigated. The relation between them and the Nambu-Goto action for a closed string is established.     

On conformal invariance of interacting fields

We study the action of the conformal algebra on interacting fields. On a certain set of states the algebra is integrated to projective representations ofSU(2,2). These representations are shown to be equivalent to the representations of the interpolated discrete series ofSU(2,2). Using this result we give a formula for the two-point Wightman function for arbitrary spin and dimension of the field. Finally we discuss the limit when the dimension tends to the canonical value.     

Essential points of conformal vector fields

An essential point of a conformal vector field ξ$\xi$ on a conformal manifold (M,c)$(M,c)$ is a point around which the local flow of ξ$\xi$ preserves no metric in the conformal class c$c$. It is well-known that a conformal vector field vanishes at each essential point. In this note we show that essential points are isolated. This is a generalization to higher dimensions of the fact that the zeros of a holomorphic function are isolated. As an application, we show that every connected component of the zero set of a conformal vector field is totally umbilical.